The Automorphism Groups of BCH Codes and of Some Affine-Invariant Codes Over Extension Fields

Affine-invariant codes are extended cyclic codes of length p ^m invariant under the affine-group acting on . This class of codes includes codes of great interest such as extended narrow-sense BCH codes. In recent papers, we classified the automorphism groups of affine-invariant codes berg, bech1. We derive here new results, especially when the alphabet field is an extension field, by expanding our previous tools. In particular we complete our results on BCH codes, giving the automorphism groups of extended narrow-sense BCH codes defined over any extension field.     

Codes from Affine Permutation Groups

It is shown that A(22,10) 50, A(23,10) 76, A(25,10) 166, A(26,10) 270, A(29,10) 1460, and A(28,12) 178, where A(n,d) denotes the maximum cardinality of a binary code of length n and minimum Hamming distance d. The constructed codes are invariant under permutations of some affine (or closely related) permutation group and have been found using computer search.     

不充当自同构群的有限群

设ｐ为奇素数，本文将用一些新的技巧来证明，当Ｐ是阶小于Ｐ＾１１的交换Ｐ－群时，自同构群方程Ａｕｔ（Ｘ）＝Ｐ无解。这个结果使ＭａｃｈＨａｌｅ在１９８３年的工作得到了突破，并且我们所给的方法具有广泛性。     

On Orbits of Automorphism Groups

If G is a finite group and if A is a group of automorphisms of G whose fixed point subgroup is C _G (A) then every subgroup F of C _G (A) acts on the set of orbits of A in G. The peculiarities of this action are used here to derive several results on the number of orbits of A in an economical manner.Original Russian Text Copyright © 2005 Deaconescu M. and Walls G. L.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 533–537, May–June, 2005.     

利用有限仿射几何构作Cartesian认证码

设 0 相似文献   

Matrix Groups Related to the Quaternion Group and Spherical Orbit Codes

We consider a finite matrix group with 3⁴· 2¹⁶ elements, which is a subgroup of the infinite group , where is the regular representation of the quaternion group and C is a matrix that transforms the regular representation Q to its cellwise-diagonal form. There is a number of ways to define the matrix C. Our aim is to make the group similar in a certain sense to a finite group. The eventual choice of an appropriate matrix C done heuristically. We study the structure of the group and use this group to construct spherical orbit codes on the unit Euclidean sphere in R⁸. These codes have code distance less than 1. One of them has 3²· 2⁸ = 2304 elements and its squared Euclidean code distance is 0.293. Communicated by: V. A. Zinoviev     

On Minimal Permutation Representations of Classical Simple Groups

We describe minimal permutation representations, i.e., faithful permutation representations of least degree, of classical finite simple groups as groups of automorphisms of simple Lie algebras.     

充当自同构群的有限群的几点注记

设p为奇素数,本文给出了中心循环中心商的阶小于p^5的有限p-群的完全分类并且给出它们中无对合自同构的群的自同构群的阶。由此,我们找到了能作为有限群自同构群的p^mq^n阶群和p^n阶群,统一和推广了Curran在1988年和Caranti与Scoppola在1990年的文章的所有结果。     

若干家族p-群的自同构群的阶

本文确定了Rodney James文章中P．Hall isoclinism族第一家族到第十家族所有群的自同构群的阶,并纠正了Rodney James文章里的某些错误．     

On the Orbit Sizes of Permutation Groups on the Power Set

We study the action of finite permutation groups of odd order on the power set of a set on which they act naturally, and establish a theorem that guarantees the existence of a lot of distinct orbit sizes in this action.1991 Mathematics Subject Classification: 20B05     

关于有限Abel p-群的自同构群

从有限Abel p-群P的型不变量出发,给出了其自同构群AutP的阶的计算公式,并利用|AutP|的计算公式得到了下面3个结果:1.由有限Abel p-群的型不变量的两种变换得到了其自同构群的阶的变化规律;2.用群的阶、秩、幂指数三个量界定了有限Abel p-群的自同构的阶;3.对部分Frattini子群为p阶群的有限p-群,确定了其自同构群的阶何时达到最小值和最大值.     

OD-Characterization of Some Projective Special Linear Groups Over the Binary Field and Their Automorphism Groups

The Gruenberg–Kegel graph GK(G) = (V _G , E _G ) of a finite group G is a simple graph with vertex set V _G  = π(G), the set of all primes dividing the order of G, and such that two distinct vertices p and q are joined by an edge, {p, q} ∈ E _G , if G contains an element of order pq. The degree deg _G (p) of a vertex p ∈ V _G is the number of edges incident to p. In the case when π(G) = {p ₁, p ₂,…, p _h } with p ₁ < p ₂ < … <p _h , we consider the h-tuple D(G) = (deg _G (p ₁), deg _G (p ₂),…, deg _G (p _h )), which is called the degree pattern of G. The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying condition (|H|, D(H)) = (|G|, D(G)). Especially, a 1-fold OD-characterizable group is simply called OD-characterizable. In this paper, we prove that the simple groups L ₁₀(2) and L ₁₁(2) are OD-characterizable. It is also shown that automorphism groups Aut(L _p (2)) and Aut(L _p+1(2)), where 2 ^p  ? 1 is a Mersenne prime, are OD-characterizable. Finally, a list of finite (simple) groups which are presently known to be k-fold OD-characterizable, for certain values of k, is presented.     

两类重要p-群及其自同构群的阶

In the paper we obtain two infinite classes of p-groups,calculate the orders of their automorphism groups and correct a mistake(perhaps misprinted)of Rodney James'paper in 1980.     

Recognizing the Automorphism Groups of Mathieu Groups

It is a well-known fact that characters of a finite group can give important information about the structure of the group. It was also proved by the third author that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost-simple group by using less information of its character table, and successfully characterize the automorphism groups of Mathieu groups by their orders and at most two irreducible character degrees of their character tables.     

自同构群阶为prq~2的有限群

本文给出满足|Aut(G)|=prq~2,q>2的有限群 G的完全分类。     

On Characteristic Polynomials of Formal Groups over Finite Fields

Generalizing the results of Serre, Hill and Koch, we give some classification theorems of higher dimensional simple formal groups over finite fields. A relation between endomorphism rings of formal groups over ?_p and characteristic polynomials of their reductions mod p is studied. A condition of existence of formal groups over ?_p with complex multiplication is given. Some formal groups over ?_p are also constructed.     

On Some Automorphisms of Orthogonal Groups in Odd Characteristic

In this paper, it is proved that the simple orthogonal groups O _2n+1(q) and O _2n ^± (q) (where q is odd) cannot be automorphism groups of finite left distributive quasigroups. This is a particular case of the conjecture stating that the automorphism group of a left distributive quasigroup is solvable. To complete the proof of the conjecture, one must test all finite groups.     

On the Characterizability of the Automorphism Groups of Sporadic Simple Groups by Their Element Orders

For G a finite group, π _e (G) denotes the set of orders of elements in G. If Ω is a subset of the set of natural numbers, h(Ω) stands for the number of isomorphism classes of finite groups with the same set Ω of element orders. We say that G is k-distinguishable if h(π _e (G)) = k < ∞, otherwise G is called non-distinguishable. Usually, a 1-distinguishable group is called a characterizable group. It is shown that if M is a sporadic simple group different from M ₁₂, M ₂₂, J ₂, He, Suz, M ^c L and O′N, then Aut(M) is characterizable by its element orders. It is also proved that if M is isomorphic to M ₁₂, M ₂₂, He, Suz or O′N, then h(π _e (Aut(M))) ∈¸ {1,∞}.     

On Ordinary Words of Standard Reed-Solomon Codes over Finite Fields

Reed-Solomon codes are widely used to establish a reliable channel to transmit information in digital communication which has a strong error correction capability and a variety of efficient decoding algorithm.Usually we use the maximum likelihood decoding(MLD) algorithm in the decoding process of Reed-Solomon codes.MLD algorithm relies on determining the error distance of received word.Dür,Guruswami,Wan,Li,Hong,Wu,Yue and Zhu et al.got some results on the error distance.For the Reed-Solomon code C,the received word u is called an ordinary word of C if the error distance d(u,C) =n-deg u(x) with u(x) being the Lagrange interpolation polynomial of u.We introduce a new method of studying the ordinary words.In fact,we make use of the result obtained by Y.C.Xu and S.F.Hong on the decomposition of certain polynomials over the finite field to determine all the ordinary words of the standard Reed-Solomon codes over the finite field of q elements.This completely answers an open problem raised by Li and Wan in[On the subset sum problem over finite fields,Finite Fields Appl.14 (2008) 911-929].